Robert, Frédéric; Vétois, Jérôme Sign-changing blow-up for scalar curvature type equations. (English) Zbl 1277.58010 Commun. Partial Differ. Equations 38, No. 7-9, 1437-1465 (2013). Summary: Given \((M,g)\) a compact Riemannian manifold of dimension \(n\geq 3\), we are interested in the existence of blowing-up sign-changing families \((u_\varepsilon)_{\varepsilon>0}\in C^{2,\theta}(M)\), \(0\in(0,1)\), of solutions to \[ \Delta_g u_\varepsilon+hu_\varepsilon=| u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon\text{ in }M, \] where \(\Delta_g:=\mathrm{div}_g(\nabla)\) and \(h\in C^{0,\theta}(M)\) is a potential. Assuming the existence of a nondegenerate solution to the limiting equation (which is a generic assumption), we prove that such families exist in two main cases: in small dimension \(n\in\{3,4,5,6\}\) for any potential \(h\) or in dimension \(3\leq n\leq 9\) when \(h\equiv\frac{n-2}{4(n-1)}\mathrm{Scal}_g\). These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of O. Druet [J. Differ. Geom. 63, No. 3, 399–473 (2003; Zbl 1070.53017)] and M. A. Khuri et al. [J. Differ. Geom. 81, No. 1, 143–196 (2009; Zbl 1162.53029)]. Cited in 18 Documents MSC: 58J05 Elliptic equations on manifolds, general theory 35J35 Variational methods for higher-order elliptic equations 35J60 Nonlinear elliptic equations 35B44 Blow-up in context of PDEs 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) Keywords:blow-up; conformal invariance; nonlinear elliptic equations Citations:Zbl 1070.53017; Zbl 1162.53029 PDFBibTeX XMLCite \textit{F. Robert} and \textit{J. Vétois}, Commun. Partial Differ. Equations 38, No. 7--9, 1437--1465 (2013; Zbl 1277.58010) Full Text: DOI arXiv References: [1] Ambrosetti A., Perturbation Methods and Semilinear Elliptic Problems on \(\mathbb{R}\) n 240 (2006) · Zbl 1115.35004 [2] Aubin T., J. Diff. Eqs. 11 pp 573– (1976) · Zbl 0371.46011 · doi:10.4310/jdg/1214433725 [3] Berger M., Le Spectre D’une Variété Riemannienne 194 (1971) · doi:10.1007/BFb0064643 [4] DOI: 10.1016/0022-1236(91)90099-Q · Zbl 0755.46014 · doi:10.1016/0022-1236(91)90099-Q [5] DOI: 10.1090/S0894-0347-07-00575-9 · Zbl 1206.53041 · doi:10.1090/S0894-0347-07-00575-9 [6] Brendle S., J. Diff. Geom. 81 pp 225– (2009) · Zbl 1166.53025 · doi:10.4310/jdg/1231856261 [7] Chen W., J. Diff, Eqs. [8] DOI: 10.1016/j.jde.2011.03.008 · Zbl 1233.35008 · doi:10.1016/j.jde.2011.03.008 [9] del Pino M., Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) [10] DOI: 10.1007/BF01209398 · Zbl 0608.35017 · doi:10.1007/BF01209398 [11] Druet O., J. Diff. Geom. 63 pp 399– (2003) · Zbl 1070.53017 · doi:10.4310/jdg/1090426771 [12] DOI: 10.1155/S1073792804133278 · Zbl 1085.53029 · doi:10.1155/S1073792804133278 [13] DOI: 10.2140/apde.2009.2.305 · Zbl 1208.58025 · doi:10.2140/apde.2009.2.305 [14] Druet O., Blow-Up Theory for Elliptic PDEs in Riemannian Geometry 45 (2004) · Zbl 1059.58017 [15] DOI: 10.1016/j.jfa.2009.07.004 · Zbl 1183.58018 · doi:10.1016/j.jfa.2009.07.004 [16] Guo Y., J. Diff. Equations [17] Hebey E., Math. Res. Lett. [18] Khuri M.A., J. Diff. Geom. 81 pp 143– (2009) · Zbl 1162.53029 · doi:10.4310/jdg/1228400630 [19] Kazdan J.L., J. Diff. Geom. 10 pp 113– (1975) · Zbl 0296.53037 · doi:10.4310/jdg/1214432678 [20] DOI: 10.1090/S0273-0979-1987-15514-5 · Zbl 0633.53062 · doi:10.1090/S0273-0979-1987-15514-5 [21] DOI: 10.1007/s00526-003-0224-y · Zbl 1078.32026 · doi:10.1007/s00526-003-0224-y [22] DOI: 10.1007/s00526-004-0320-7 · Zbl 1229.35071 · doi:10.1007/s00526-004-0320-7 [23] DOI: 10.1016/j.jfa.2006.11.010 · Zbl 1229.35072 · doi:10.1016/j.jfa.2006.11.010 [24] DOI: 10.1142/S021919979900002X · Zbl 0973.53029 · doi:10.1142/S021919979900002X [25] Marques F.C., J. Diff. Geom. 71 pp 315– (2005) · Zbl 1101.53019 · doi:10.4310/jdg/1143651772 [26] DOI: 10.1512/iumj.2009.58.3633 · Zbl 1173.58008 · doi:10.1512/iumj.2009.58.3633 [27] Obata M., J. Diff. Geom. 6 pp 247– (1971) · Zbl 0236.53042 · doi:10.4310/jdg/1214430407 [28] Pistoia A., J. Diff. Eqs. [29] DOI: 10.1016/0022-1236(90)90002-3 · Zbl 0786.35059 · doi:10.1016/0022-1236(90)90002-3 [30] Rodemich E., The Sobolev Inequalities with Best Possible Constants (1966) [31] Schoen , R.M. ( 1989 ). Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In:Topics in Calculus of Variations, (Montecatini Terme, 1987).Lecture Notes in Mathematics, vol. 1365. Berlin: Springer, pp. 120–154 . · doi:10.1007/BFb0089180 [32] Schoen R.M., Differential Geometry 52 pp 311– (1991) [33] DOI: 10.1007/BF01174186 · Zbl 0535.35025 · doi:10.1007/BF01174186 [34] Talenti G., Ann. Mat. Pura Appl. 110 pp 353– · Zbl 0353.46018 · doi:10.1007/BF02418013 [35] DOI: 10.1142/S0129167X0700445X · Zbl 1148.35026 · doi:10.1142/S0129167X0700445X [36] Wei J.-C., Ann. Sc. Norm. Super. Pisa Cl. Sci. 9 pp 423– (2010) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.